Step 1: Understanding the Concept:
We have an implicit equation relating \( x \) and \( y \). We need to find the derivative \( \frac{dy}{dx} \) using implicit differentiation.
Step 2: Key Formula or Approach:
Differentiate both sides of the equation with respect to \( x \).
Remember to use the chain rule for terms involving \( y \): \( \frac{d}{dx}(y^n) = n y^{n-1} \cdot \frac{dy}{dx} \).
Step 3: Detailed Explanation:
Given equation:
\[ x^{2/5} + y^{2/5} = a^{2/5} \]
Differentiate with respect to \( x \):
\[ \frac{d}{dx}(x^{2/5}) + \frac{d}{dx}(y^{2/5}) = \frac{d}{dx}(a^{2/5}) \]
Since \( a \) is a constant, its derivative is zero.
\[ \frac{2}{5}x^{(2/5) - 1} + \frac{2}{5}y^{(2/5) - 1} \cdot \frac{dy}{dx} = 0 \]
\[ \frac{2}{5}x^{-3/5} + \frac{2}{5}y^{-3/5} \frac{dy}{dx} = 0 \]
Divide the entire equation by \( \frac{2}{5} \):
\[ x^{-3/5} + y^{-3/5} \frac{dy}{dx} = 0 \]
Now, isolate \( \frac{dy}{dx} \):
\[ y^{-3/5} \frac{dy}{dx} = -x^{-3/5} \]
\[ \frac{dy}{dx} = -\frac{x^{-3/5}}{y^{-3/5}} \]
Using the property of exponents \( a^{-n} = \frac{1}{a^n} \), we can flip the fraction:
\[ \frac{dy}{dx} = - \frac{y^{3/5}}{x^{3/5}} \]
\[ \frac{dy}{dx} = - \left(\frac{y}{x}\right)^{3/5} \]
This can be written in radical form:
\[ \frac{dy}{dx} = - \sqrt[5]{\left(\frac{y}{x}\right)^3} \]
Step 4: Final Answer:
The derivative is \( -\sqrt[5]{\left(\frac{y}{x}\right)^3} \).